Root separation for reducible integer polynomials

(d) ≥ (d +1)/3, for every d ≥ 2.Except those from [2], the above results are obtained by presenting ex-plicit families of (irreducible) polynomials of degree d whose roots are closeenough. The ingenious proof in [2] does not give any explicit example ofsuch polynomials, but shows that algebraic numbers of degree d with a closeconjugate form a ‘highly dense’ subset in the real line.The aim of the present note is to improve all known lower bounds fore